Marginal likelihoods are the currency of model comparison in a Bayesian
framework.
This differs from the frequentist approach to model
choice, which is based on comparing the maximum probability or density of the
data under two models either using a likelihood ratio test or some
information-theoretic criterion.
Because adding a parameter (dimension) to a model will always ensure
a maximum likelihood at least as large as without the parameter, some
penalty must be imposed when parameters are added.
How large this penalty should be is not easy to define, which has
led to many different possible criteria, e.g.,
the Akaike information criterion [1]},
second-order AIC [2]}, [3]},
and Bayesian information criterion
[4]}.
